What is the ruling on the prayer of one who prayed while having a minor ritual impurity, knowing of it, and was unable to renew his ablution until after the prayer, and what is the expiation for this sin?

It is not permissible for a Muslim to pray without ablution (wudu') or dry ablution (tayammum). Whoever does so intentionally, knowing about the ritual impurity (hadath) and the prohibition of prayer with it, has committed a grave sin that necessitates repentance. That prayer is not valid for him. However, he does not become an disbeliever by this, unless he deems it permissible. She must repent and increase in good deeds.